Escape rates and Hausdorff dimension of Julia sets (Q686071)

A periodic orbit formalism is used to compute escape rate and Hausdorff dimension for the Julia sets of the one parameter family of maps \(z \Rightarrow z^ 2 + c\) in a neighborhood of \(c = 0\). The method is based on \(\Omega(z,D) = \sum^ \infty_{n = 1}z^ n\sum_ P|\Lambda_ P|^{-D}\), where \(\Lambda_ P\) is the derivative of the \(n\)-th iterate of the function at the point \(P\) of the complex plane, periodic after \(n\) iterations. \(\Omega(z,D)\) can be viewed as a logarithmic derivative of a zeta function and so the escape rate and Hausdorff dimension is calculated as certain zeros of this zeta function.